The origin of the energy-momentum conservation law
aa r X i v : . [ h e p - t h ] J u l The origin of the energy-momentum conservation law
Andrew E Chubykalo † , Augusto Espinoza † , and B P Kosyakov ‡§ July 14, 2017 † Escuela de F´ısica, Universidad Aut´onoma de Zacatecas, Apartado Postal C-580 Zacatecas98068, Zacatecas, Mexico ‡ Russian Federal Nuclear Center, Sarov, 607189 Nizhni˘ı Novgorod Region, Russia § Moscow Institute of Physics & Technology, Dolgoprudni˘ı, 141700 Moscow Region, Russia
E-mail: achubykalo@yahoo . com . mx, drespinozag@yahoo . com . mx, kosyakov . boris@gmail . com Abstract
The interplay between the action–reaction principle and the energy-momentumconservation law is revealed by the examples of the Maxwell–Lorentz and Yang–Mills–Wong theories, and general relativity. These two statements are shown to beequivalent in the sense that both hold or fail together. Their mutual agreement isdemonstrated most clearly in the self-interaction problem by taking account of therearrangement of degrees of freedom appearing in the action of the Maxwell–Lorentzand Yang–Mills–Wong theories. The failure of energy-momentum conservation ingeneral relativity is attributed to the fact that this theory allows solutions havingnontrivial topologies. The total energy and momentum of a system with nontrivialtopological content prove to be ambiguous, coordinatization-dependent quantities.For example, the energy of a Schwarzschild black hole may take any positive valuegreater than, or equal to, the mass of the body whose collapse is responsible forarising this black hole. We draw the analogy to the paradoxial Banach–Tarskitheorem; the measure becomes a poorly defined concept if initial three-dimensionalbounded sets are rearranged in topologically nontrivial ways through the action offree non-Abelian isometry groups.
Keywords: action–reaction, translation invariance, energy and momentum conservation,rearrangement of initial degrees of freedom
By summing the basic advances in physics of the 19th century, Max Planck placed strongemphasis on the action–reaction principle as the rationale of momentum conservation[1]. On the other hand, following Noether’s first theorem [2], we recognize that anydynamical system exhibits momentum conservation if the action of this system is invariantunder space translations, in other words, the momentum conservation law stems fromhomogeneity of space.In nonrelativistic mechanics, Newton’s third law is consistent with the requirement oftranslation invariance. Indeed, the forces exerted on particles in an isolated two-particle1ystem are on the same line, equal, and oppositely directed when the potential energyassumes the form U ( z − z ), where z and z are coordinates of these particles. However,this law is no longer valid in relativistic mechanics where the influence of one particleon another propagates at a finite speed, and the response arises with some retardation.Furthermore, energy and momentum are fused into energy-momentum whose conservationis attributed to homogeneity of Minkowski spacetime. So the Planck’s insight into thereason for momentum conservation is gradually fading from the collective consciousnessof theoretical physics.Meanwhile there is one exceptional case, namely contact interactions, in which oneparticle acts on another and experiences back reaction at the same point, as exemplifiedby collisions and decays of pointlike particles. This form of relativistic interactions respectsboth Newton’s third law and energy-momentum conservation, suggesting to consider theaction–reaction principle in a broader sense and extend it to cover local interactions inclassical field theories. The most familiar example can be found in the Maxwell–Lorentzelectrodynamics in which the role of the electric charge e is twofold: e acts as both couplingbetween the point particle carrying this charge and electromagnetic field and the strengthof the delta-function source of electromagnetic field.To gain a clearer view of whether the action–reaction principle has a direct bearingon energy–momentum conservation, one should invoke the self-interaction problem. Thisissue is studied in Sects. 2 and 3 by the examples of Maxwell–Lorentz electrodynamicsand Yang–Mills–Wong theory.Turning to general relativity in Sect. 4, we conclude that both action–reaction principleand energy–momentum conservation cease to be true. The absence of energy-momentumconservation from this theory is due to the fact that the equation of gravitational fieldallows solutions which represent spacetime manifolds with nontrivial topology. Energy andmomentum may thus become poorly defined concepts in general relativity. It transpiresthat the total energy of a Schwarzschild black hole may take any positive value greaterthan, or equal to, the mass of the collapsed body in different coordinatizations. Thesituation closely resembles that in the paradoxial Banach–Tarski theorem. We sketch thebroad outline of this theorem and its potential relevance to the problem of poorly definedmeasure for total energy and momentum in Sects. 4 and 5. The rearrangement of degreesof freedom appearing in the action and its role in facilitating the integral quantities tobecome well-defined is discussed in Sect. 5.We follow the notation used in [3]. In Sects. 2, 3, and 5, in which our attention isrestricted essentially to the picture in Minkowski spacetime, we adopt the mainly negativesignature (+ − −− ) convenient to the description of world lines. In Sect. 4, we proceedfrom the idea of pseudo-Riemannian spacetime, and use the mainly positive signature( − + ++), which is particularly adapted to the description of 3-dimensional surfaces. Weput the speed of light equal to unity throughout. The action S = − Z d x (cid:18) π F µν F µν + j µ A µ (cid:19) − m Z dτ p ˙ z µ ˙ z µ (1)2ncodes the dynamics of a single charged particle interacting with electromagnetic field.Here, j µ ( x ) = e Z ∞−∞ dτ ˙ z µ ( τ ) δ [ x − z ( τ )] (2)is the current density produced by the particle moving along a smooth timelike world line z µ ( τ ) and carrying the charge e , and m is the mechanical mass of this bare particle.A closed system of this kind enjoys the property of translational invariance whichaffords energy-momentum conservation through the famous Noether argument.The comparison of the source, Eq. (2), in the field equation E µ = ∂ ν F µν + 4 πj µ = 0 (3)with the Lorentz force in the equation of motion for this charged particle ε λ = m ¨ z λ − e ˙ z µ F λµ = 0 , (4)where the dot stands for the derivative with respect to the proper time s of the particle,shows that both are scaled by the same parameter e . This fact is consistent with theaction–reaction principle: e measures both variation of the particle state for a given fieldstate and variation of the field state for a given particle state.Does this statement bear on energy-momentum conservation? To answer this question,we turn to the self-interaction problem. Naively, this problem is about interfacing thebare particle and electromagnetic field on the world line, which will hopefully reveal localenergy-momentum balance of this contact interaction. We are therefore to address asimultaneous solution of equations (3) and (4). To see this, consider the Noether identity ∂ µ T λµ = 14 π E µ F λµ + Z ∞−∞ ds ε λ ( z ) δ [ x − z ( s )] , (5)where T µν is the total metric stress-energy tensor of this system, T µν = 2 √− g δSδg µν = Θ µν + t µν , (6)Θ µν = 14 π (cid:18) F µα F να + 14 η µν F αβ F αβ (cid:19) , (7) t µν = m Z ∞−∞ ds ˙ z µ ( s ) ˙ z ν ( s ) δ [ x − z ( s )] , (8)and E µ and ε λ are, respectively, the left-hand sides of equations (3) and (4). It followsfrom (5) that E µ = 0 and ε λ = 0 imply ∂ µ T λµ = 0, that is, the equation of motion for abare particle (4), in which an appropriate solution to the field equation (3) has been used,is equivalent to the local conservation law for the total stress-energy tensor.Imposing the retarded boundary condition, we obtain a solution to equation (3) in theLi´enard–Wiechert form, F µν ret = eρ ( R µ V ν − R ν V µ ) , (9) V µ = [1 − ( R · ¨ z )] ˙ z µ ρ + ¨ z µ , (10)3 µ = x µ − z µ ( s ret ) is a lightlike vector drawn from a point z µ ( s ret ) on the world line, wherethe signal was emitted, to the point x µ , where the signal was received, and ρ = R · ˙ z is thespatial distance between x µ and z µ ( s ret ) in the instantaneously comoving Lorentz framein which the charge is at rest at the retarded instant s ret . The field (9)–(10) is singular onthe world line. Substituting it into (4) results in a divergent expression. This divergenceis a manifestation of infinite self-interaction: the charged bare particle experiences its ownelectromagnetic field which is infinite at the point of origin.A possible cure for this difficulty is to regularize the Li´enard–Wiechert field F µν ret in asmall vicinity of the world line. Take, for example, the field as a function of two variables F µν ret ( x ; z ( s ret )) and continue it analytically from null intervals between the observationpoints x µ and the retarded points z µ ( s ret ) to timelike intervals that result from assigning x µ = z µ ( s ret + ǫ ) and keeping the second variable z µ ( s ret ) fixed [4]. A crucial step inremoving the regularization is to change m to a function of regularization, m ( ǫ ), add itto the divergent term e / ǫ , and assume that m = lim ǫ → (cid:20) m ( ǫ ) + e ǫ (cid:21) (11)is finite and positive. Then the divergence disappears, and we arrive at the Lorentz–Diracequation [5] Λ µ = m ¨ z µ − e (cid:0) ... z µ + ˙ z µ ¨ z (cid:1) − f µ ext = 0 , (12)where f µ ext = e ˙ z ν F µν ext is an external four-force, with F µν ext being a free electromagnetic field.Is it possible to regard (12) as the desired equation of local energy-momentum balance?Based on the wide-spread belief that the Abraham termΓ µ = 23 e (cid:0) ... z µ + ˙ z µ ¨ z (cid:1) (13)is the radiation reaction four-force , one would give a negative answer to this question.This is because the radiating particle feels a recoil equal to the negative of the Larmoremission rate ˙ P µ = − e ˙ z µ ¨ z . (14)However, − ˙ P µ cannot be considered as a four-force because it is not orthogonal to ˙ z µ .On the other hand, Γ µ is orthogonal to ˙ z µ , but it differs from the anticipated recoil bythe so-called Schott term e ... z µ . Although the energy stored in the Schott term can beattributed to a reversible form of emission and absorption of field energy [5], its actualrole appears mysterious.Furthermore, the general solution to equation (12) with f µ ext = 0 is˙ z µ ( s ) = e µ cosh( α + w τ e s/τ ) + e µ sinh( α + w τ e s/τ ) , (15)where e µ and e µ are constant vectors such that e · e = 0, e = − e = 1, τ = 2 e / m , α and w are arbitrary constants. The solution (15) is an embarrassing feature of theLorentz–Dirac equation: a free charged particle moving along this world line continuallyaccelerates, ¨ z ( s ) = − w exp (2 s/τ ) , (16)4nd continually radiates. This self-acceleration seems contrary to the energy-momentumconservation law even though this law is assured by translational invariance of the action.These paradoxial results signal that self-interaction is a subtle issue whose treatmentrequires further refinements of the conceptual basis. A plausible assumption is that theextremization of the action, subject to the retarded condition, may result in unstablemodes, which culminates in rearranging the initial degrees of freedom [3]. The action(1) is expressed in terms of mechanical variables z µ ( τ ) describing world lines of a barecharged particle and the electromagnetic vector potential A µ ( x ). The rearrangement ofthese degrees of freedom yields new dynamically independent entities, a dressed chargedparticle and radiation .We begin with the local conservation law for the total stress-energy tensor ∂ λ T λµ = 0 . (17)Recall that taking the local conservation law (17), as the starting point in the self-energyanalysis, is as good as that of simultaneous solution of dynamical equations (3) and (4).Substituting the general solution of the field equation (3), F µν = F µν ret + F µν ext , into (7) givesΘ µν = − e πρ (cid:20) V R µ R ν − ( R µ V ν + R ν V µ ) + 12 η µν (cid:21) + Θ µν mix , (18)where the first term results from the self-field (9)–(10), and the second term containsmixed contributions of the self-field and free field. The first term splits into two partsΘ µν bound + Θ µν rad , whereΘ µν bound = − e πρ (cid:20) R µ R ν ρ (1 − R · ¨ z ) − ( R µ V ν + R ν V µ ) + 12 η µν (cid:21) , (19)Θ µν rad = − e πρ (cid:20) ¨ z + 1 ρ (¨ z · R ) (cid:21) R µ R ν . (20)The following local conservation laws hold off the world line [7]: ∂ µ Θ µν bound = 0 , ∂ µ Θ µν rad = 0 , ∂ µ Θ µν mix = 0 . (21)A natural interpretation of (21) is that Θ µν bound , Θ µν rad , and Θ µν mix are dynamically independent outside the world line [7]. There is no other decomposition of Θ µν into parts which maybe recognized as dynamically independent.Since Θ µν rad and Θ µν mix behave like ρ − near the world line, they are integrable over athree-dimensional spacelike surface Σ, and, in view of (21), the surface of integrationmay be deformed from Σ to more geometrically motivated surfaces. It is convenient tosubstitute Σ by a tube T ǫ of infinitesimal radius ǫ enclosing the world line to obtain P µ = Z Σ dσ λ Θ λµ rad = lim ǫ → Z T ǫ dσ λ Θ λµ rad = − e Z s −∞ dτ ¨ z ( τ ) ˙ z µ ( τ ) (22)and ℘ µ = Z Σ dσ λ Θ λµ mix = lim ǫ → Z T ǫ dσ λ Θ λµ mix = − e Z s −∞ dτ F µν ext ( z ) ˙ z ν ( τ ) . (23) For other paradoxes related to self-interaction in the Maxwell–Lorentz theory see, e. g., [6]. µ represents the four-momentum radiated by the charge e during the whole pasthistory prior to the instant s . Indeed: (i) Θ µν rad is a dynamically independent part of Θ µν ;(ii) Θ µν rad moves away from the charged particle with the speed of light, more precisely, Θ µν rad propagates along the future light cone C + drawn from the emission point, Θ µν rad R ν = 0;(iii) the flux of Θ µν rad goes as ρ − implying that the same amount of energy-momentumflows through spheres of different radii. Differentiating (22) with respect to s gives theLarmor four-momentum emitted by the accelerated charge per unit proper time, Eq. (14).As for ℘ µ , it is the four-momentum extracted from the free field F µν ext ( x ) during thewhole past history up to the instant s .By contrast, Θ µν bound contains singularities ρ − and ρ − which are not integrable. Hence,an appropriate regularization is necessary. For example, employing a Lorentz-invariantcutoff prescription [3], one finds P µ bound = Reg ǫ Z Σ dσ λ Θ λµ bound = e ǫ ˙ z µ − e ¨ z µ , (24)where ǫ is the cutoff parameter which must go to zero in the end of calculations. Sincethe flux of Θ µν bound through C + is nonzero, Θ µν bound R ν = 0, Θ µν bound propagates slower thanlight. Unlike Θ µν rad , which detaches from the source, Θ µν bound remains bound to the source[7]. In other words, the source carries the four-momentum P µ bound along with its motion.From (24) follows that the measure dσ λ Θ λµ bound is ill-defined. However, observing that p µ = Z Σ dσ λ t λµ = m ˙ z µ , (25)one may render m a singular function of ǫ , m ( ǫ ), add Eqs. (24) and (25) up, and carryout the renormalization of mass, Eq. (11), to complete the definition of the measureReg ǫ dσ λ (cid:16) Θ λµ bound + t λµ (cid:17) in the limit ǫ →
0, and eventually arrive at p µ = lim ǫ → Reg ǫ Z Σ dσ λ (cid:16) Θ λµ bound + t λµ (cid:17) = m ˙ z µ − e ¨ z µ . (26)This four-momentum, originally deduced in [7], is to be attributed to the dressed particle.We now integrate (17) over a domain of spacetime bounded by two spacelike surfacesΣ ′ and Σ ′′ , separated by a small timelike interval, with both normals directed towards thefuture, and a tube T R of large radius R . Applying the Gauss–Ostrogradskiˇı theorem, weobtain : (cid:18)Z Σ ′′ − Z Σ ′ + Z T R (cid:19) dσ µ (cid:0) Θ λµ + t λµ (cid:1) = (cid:26) lim ǫ → (cid:20) m ( ǫ ) + e ǫ (cid:21) ˙ z λ − e ¨ z λ (cid:27) ∆ s − Z s +∆ ss dτ (cid:20) e ¨ z ( τ ) ˙ z λ ( τ ) + eF λµ ext ( z ) ˙ z µ ( τ ) (cid:21) = 0 , or, in a concise form [3], ∆ p λ + ∆ P λ + ∆ ℘ λ = 0 . (27)Evidently (27) is identical to the Lorentz–Dirac equation (12). We assume that F λµ ext ( x ) disappears at spatial infinity. Therefore, the only term contributing to theintegral over T R is Θ µν rad . Taking into account the second equation of (21), the integral of Θ µν rad over T R can be converted into the integral over T ǫ , so that the upshot is given by Eq. (22).
6n the other hand, (27) is the desired energy-momentum balance : the four-momentum∆ ℘ λ = − eF λµ ext ˙ z µ ∆ s which is extracted from the external field F λµ ext during the period oftime ∆ s is distributed between the four-momentum of the dressed particle ∆ p λ and thefour-momentum carried away by radiation ∆ P λ .Of particular interest is the case F λµ ext = 0,∆ p λ = − ∆ P λ . (28)It immediately follows that the rate of change of the energy-momentum of a dressedparticle, ˙ p λ , is equal to the negative of the Larmor emission rate, − ˙ P λ . Here, two remarksare in order. First, − ∆ P λ is a mere four-momentum (rather than four-force), and hencethe fact that ˙ P λ is not orthogonal to ˙ z λ presents no special problem. Second, the energyof a dressed particle is indefinite , p = mγ (cid:0) − τ γ a · v (cid:1) , (29)where γ is the Lorentz factor γ = (1 − v ) − / . Therefore, increasing | v | need not beaccomplished by increasing p . For instance, one may readily check that the energy of adressed particle executing a self-accelerated motion (15) steadily decreases, which exactlycompensates the increase in energy of the electromagnetic field emitted . A possible interpretation of Eq. (28) is that the dressed particle experiences a jet thrust in responseto emitting the electromagnetic field momentum, a kind of apparent force applied to the same point inwhich the emission occurs. The fact that p is not positive definite is scarcely surprising. Recall that p µ is the sum of two vectors p µ = m ˙ z µ + P µ bound . The bound four-momentum P µ bound is a timelike future-directed vector, while thefour-momentum of a bare particle m ˙ z µ is a timelike past-directed vector because m ( ǫ ) < ǫ , as (11) suggests. Assuming that m ˙ z µ + P µ bound is a timelike vector, one recognizes that the timecomponent of this vector can have any sign. The solution (15) is usually thought of as a pathological trait of the Lorentz–Dirac equation (12)for two main reasons: (i) this solution seems incompatible with energy conservation, and (ii) thereis no experimental evidence for self-accelerated motions in the Nature. Both accusations are unjust.The fact that energy-momentum is conserved in this motion has just now been established. As to themanifestation of this phenomenon, the universe as a whole exhibiting accelerated expansion provides anexcellent potential example of a free entity (brane?) which executes exponentially accelerated motionwith the characteristic time equal to the inverse of current Hubble scale and emits gravitational radiation[8]. Why is the self-accelerated motion of charged particles never observed? It follows from (26) that p = m (cid:0) τ a (cid:1) . (30)If τ a < −
1, the dressed charged particle turns to a tachyonic state p <
0. Let the particle be moving inthe self-accelerated regime (15). Then, after a lapse of time ∆ t = − τ log τ | w | , the critical acceleration | a | = τ − is exceeded, and the four-momentum of this dressed particle becomes spacelike. The period oftime ∆ t over which a self-accelerated electron possesses timelike four-momenta is estimated at τ ∼ − s for electrons, and still shorter for more massive charged elementary particles. All primordial self-accelerated particles with such τ ’s have long been in the tachyonic state. However, we have not slightestnotion of how tachyons can be experimentally recorded [8]. Noteworthy also is that non-Galilean andGalilean regimes of motion are never interconvertible: the history of a particular dressed charged particleis decided by the asymptotic condition in the limit s → ∞ . It is then conceivable that the Galilean formof evolution, corresponding to w = 0 in Eq. (15), may well be assigned to all dressed charged particles. The Yang–Mills–Wong theory
The Yang–Mills–Wong theory describes the classical interaction of particles carrying non-Abelian charges with the corresponding Yang–Mills field [9]. A system of K such particles(thereafter called quarks) interacting with the SU( N ) Yang–Mills field is governed by theaction [10] S = − K X I =1 Z dτ I ( m I p ˙ z I · ˙ z I + N − X a =1 N X i,j =1 q aI η ∗ Ii (cid:20) δ ij ddτ I + ˙ z µI (cid:0) A aµ T a (cid:1) ij (cid:21) η jI ) − π Z d x G µνa G aµν , (31)where T a are generators of SU( N ), G aµν = ∂ µ A aν − ∂ ν A aµ + if abs A bµ A cν is the field strength, f abc are the structure constants of SU( N ) thereafter called the color gauge group.Quarks, labelled with I , possess color charges Q I in the adjoint representation ofSU( N ), Q I = Q aI T a . These quantities can be expressed in terms of the basic variables η Ij in the fundamental representation, Q I = N − X a =1 N X i,j =1 q aI η ∗ Ii ( T a ) ij η jI . (32)The Euler–Lagrange equations for η and η ∗ read˙ η i = − ( ˙ z · A a ) ( T a ) ij η j , ˙ η ∗ j = η ∗ i ( ˙ z · A a ) ( T a ) ij . (33)They can be combined into the Wong equation for the color charge evolution [9],˙ Q a = − if abc Q b ( ˙ z · A c ) . (34)It is convenient to rescale the color variables: Q → − ig Q a T a , A µ → ( i/g ) A aµ T a , where g is the Yang–Mills coupling constant. Then Eq. (34) becomes˙ Q = ig [ Q, ˙ z µ A µ ] . (35)It follows that the color charge Q shares with a top the property of precessing aroundsome axis in the color space.Varying z µ and A µ in the action (31) gives the dynamical equations for respectivelythe Yang–Mills field and quarks: D λ G λµ = 4 πg K X I =1 Z ∞−∞ dτ I Q I ( τ I ) ˙ z Iµ ( τ I ) δ [ x − z I ( τ I )] , (36) m I ¨ z λI = ˙ z Iµ tr (cid:2) Q I G λµ ( z I ) (cid:3) . (37)In contrast to the electric charge e , which is a constant, the color charge Q is adynamical variable governed by the Wong equation (35). Note, however, that the colorcharge magnitude is a constant of motion, dds | Q | = 2 ˙ Q a Q a = 0 , (38)8hich can be readily seen from (34) written in the Cartan basis in which f abc = − f bac .Furthermore, there is good reason to look for solutions of the Yang–Mills equations (36)satisfying the condition Q a ( s ) = const . (39)Abandoning this condition would pose the problem of an infinitely rapid precession of Q in view of the fact that the retarded field A µ is singular on the world line.It is clear from Eqs. (36) and (37) that the action–reaction principle holds in theYang–Mills–Wong theory. If one conceives that only a single quark is in the universe,then Q measures both the variation of the quark state for a given field state and variationof the field state for a given quark state.Again, we look at self-interaction for revealing the relation between the action–reactionprinciple and energy-momentum conservation in an explicit form. The strategy here copiesthat in the Maxwell–Lorentz electrodynamics, but has several traits associated with thefact that the field equations (36) are nonlinear.There are two kinds of retarded solutions to the Yang–Mills equations, Abelian andnon-Abelian [11]. We first turn to the simplest case that the SU(2) Yang–Mills field isgenerated by a single quark moving along an arbitrary timelike smooth world line [12].The retarded Abelian solution A µ = qT ˙ z µ ρ (40)resembles the Li´enard–Wiechert solution of the Maxwell–Lorentz electrodynamics, whereasthe retarded non-Abelian solution is given by A µ = ∓ ig T ˙ z µ ρ + iκ ( T ± iT ) R µ . (41)Here T a , ( a = 1 , , q and κ are arbitrary real nonzeroparameters.A remarkable feature of retarded non-Abelian solutions bearing on our discussion isthat the Yang–Mills equations determine not only the field, but also the color charge thatgenerates this field, as exemplified by (41). This solution admits only a single value forthe magnitude of the color charge carried by the quark [12], [13], | Q | = − g . (42)Recall that the electric charge e of any particle in the Maxwell–Lorentz electrodynamicsmay be arbitrary. The selection of a special magnitude for the color charge of the sourcetakes place also for a closed system of K quarks evolving in the non-Abelian regime [11],tr ( Q I ) = − g (cid:18) − N (cid:19) , N ≥ . (43)Clearly this feature of the non-Abelian dynamics offers no danger to the fulfilment of theaction–reaction principle.In the Abelian regime , the field equations (36) linearize , and hence, their retardedsolution shows up as that in (9)–(10). All results of the previous section are reproduced The reason for this linearization is that the color field variables are restricted to the Cartan subalgebraof the Lie algebra of the associated gauge group. Since the Lie algebra su( N ) is of rank N −
1, thereexist
N − H a forming a Cartan subalgebra of commuting matrices. e → q . The degrees of freedom appearing in the action(31) are rearranged on the extremals subject to the retarded condition to give a dressedquark and Yang–Mills radiation, closely resembling such entities in electrodynamics. Thebehavior of a dressed quark is governed by the Lorentz–Dirac equation (12), which canbe converted to the local energy-momentum balance (27).In the non-Abelian regime , the field equations (36) remain nonlinear, and superposingtheir solutions ceases to be true. Aside from the one-quark solution (41), there is needto examine K -quark solutions, K ≥
2. A consistent Yang–Mills–Wong theory can beformulated for the color gauge group SU( N ) with N ≥ K + 1 [11]. As an illustration werefer to a retarded SU(3) field due to two quarks [14], [15], A µ = ∓ ig (cid:18) H ˙ z µ ρ + g κ E ± R µ (cid:19) ∓ ig (cid:18) H ˙ z µ ρ + g κ E ± R µ (cid:19) . (44)Here, H a and E ab are generators of SU(3) in the Cartan–Weyl basis, which are expressedin terms of the Gell-Mann matrices as follows: H = 12 (cid:18) λ + λ √ (cid:19) , H = − (cid:18) λ − λ √ (cid:19) , E = 12 ( λ + iλ ) , E = 12 ( λ + iλ ) . (45) R µ = x µ − z µ ( τ ) and R µ = x µ − z µ ( τ ) are, respectively, the four-vectors drawn frompoints z µ ( τ ) and z µ ( τ ) on the world lines of quarks 1 and 2, where the signals wereemitted, to the point x µ , where the signals were received.Observing that A µ is the sum of two single-quark terms, one may wonder of howthe nonlinearity of the Yang–Mills equations is compatible with this fact. The answer issimple: two single-quark vector potentials with the fixed magnitudes of the color charges,as shown in Eq. (43), are combined in Eq. (44), but it is impossible to build solution asan arbitrary superposition of these terms. If either of them is multiplied by a coefficientdifferent from 1 and added to another, no further solution arises.Due to this feature – which is characteristic of the general K -quark case – we haveΘ µν = X I Θ µνI + X J = I Θ µνIJ ! , (46)where Θ µνI is comprised of the field generated by the I th quark, and Θ µνIJ contains mixedcontributions of the fields due to the I th and J th quarks. Furthermore, Θ µνI splits intobound and radiated parts. Every term of Eq. (46) satisfies the local conservation law ofthe type shown in Eq. (21), and hence represents a dynamically independent part of Θ µν .The stress-energy tensor Θ µν is thus similar in structure to that in the Maxwell–Lorentztheory.We now restrict our attention to a single quark of this K -quark system. For notationalconvenience, we omit the quark labelling.Using the line of reasoning developed in the previous section, and observing that thelinearly rising term of A µ does not contribute to Θ µν becausetr ( H l E ± mn ) = 0 , tr (cid:0) E ± kl E ± mn (cid:1) = 0 , (47)we arrive at the conclusion that the four-momentum of the retarded Yang–Mills fieldgenerated by the quark under study is given by P µ = P µ bound + P µ , where P µ bound and P µ are respectively the bound and radiated parts of this four-momentum.10n accelerated quark emits P µ = −
23 tr ( Q ) Z s −∞ dτ ¨ z ˙ z µ . (48)Owing to the negative norm of the color charges, Eqs. (42) and (43), the emitted energyis negative, which suggests that the quark gains, rather than loses, energy by emittingthe Yang–Mills radiation in the non-Abelian regime. An explicit calculation shows thatthis is indeed the case: ˙ P · ˙ z = 83 g (cid:18) − N (cid:19) ¨ z < . (49)This phenomenon might be interpreted as absorbing convergent waves of positive energyrather than emitting divergent waves of negative energy [13].Adding the bound part of the field four-momentum P µ bound = tr ( Q ) (cid:18) ǫ ˙ z µ −
23 ¨ z µ (cid:19) (50)to the mechanical four-momentum p µ = m ˙ z µ , and carrying out the renormalization ofmass in a way similar to (11), gives the four-momentum of a dressed quark p µ = m ( ˙ z µ + ℓ ¨ z µ ) . (51)Here, m is the renormalized mass, and ℓ = 83 mg (cid:18) − N (cid:19) (52)is a characteristic length inherent in the non-Abelian dynamics of this dressed quark.The mixed terms in Eq. (46) have integrable singularities ρ − on every world line. Theirtreatment is therefore similar to that of Θ µν mix in the Maxwell–Lorentz electrodynamics.The integration of these terms gives ℘ µ = − Z s −∞ dτ f µ ext [ z ( τ )] , (53)where the integrand is the color four-force exerted on the given quark by all other quarksat the instant τ . The explicit form of f µ ext is of no concern in the present context.We reiterate mutatis mutandis the argument of the previous section to find˙ p µ + ˙ P µ = f µ ext . (54)According to this balance equation, the four-momentum d℘ µ = − f µ ext ds extracted froman external field is used for changing the four-momentum of the dressed quark dp µ andemitting the Yang–Mills radiation four-momentum d P µ . A special feature of equation(54) is that d P is associated with emitting negative-energy waves or, what is the same –absorbing positive-energy waves.Substitution of (51), (48), and (53) into (54) gives the equation of motion for a dressedquark m (cid:2) ¨ z µ + ℓ (cid:0) ... z µ + ˙ z µ ¨ z (cid:1)(cid:3) = f µ ext , (55)11iffering from the Lorentz–Dirac equation (12) only in the overall sign of the parenthesizedterm and changing τ by ℓ . If f µ ext = 0, the general solution of equation (55) is˙ z µ ( s ) = V µ cosh( α + w ℓ e − s/ℓ ) + U µ sinh( α + w ℓ e − s/ℓ ) , (56)where V µ and U µ are constant four-vectors such that V · U = 0, V = − U = 1, and α and w are arbitrary parameters. A free quark may therefore execute a non-uniformmotion with exponentially decreasing acceleration. The world line of this self-deceleratedmotion asymptotically approaches a straight line. This situation can be interpreted inthe spirit of the action–reaction principle. Equation (54) becomes dp µ = − d P µ . (57)The free dressed quark feels the ‘reverse’ four-momentum transfer (responsible for theself-deceleration) because the phenomenon of radiating the Yang–Mills four-momentumis actually changed by that of absorbing this four-momentum. Nevertheless, Eq. (57)offers direct evidence that the action–reaction principle is equivalent to energy-momentumconservation on the world line. The action–reaction principle does not hold in the gravitational interaction described bygeneral relativity. Indeed, the coupling between a particle of mass m and the gravitationalfield is equal to m , so that the particle is governed by the geodesic equation d z λ dτ + Γ λµν dz µ dτ dz ν dτ = 0 , (58)which is mass-independent. On the other hand, the field equation R µν − Rg µν − πG N t µν = 0 (59)with the delta-function source t µν ( x ) = m Z ∞−∞ dτ ˙ z µ ( τ ) ˙ z ν ( τ ) δ [ x − z ( τ )] (60)shows that the greater is m , the stronger is the generated gravitational field. The influenceof particles on the state of the gravitational field is different for different m , even thoughthe gravitational field exerts on every particle in a uniform way, no matter what is m .This is contrary to the action–reaction principle.Does this violation of the action–reaction principle imply that the energy–momentumconservation law is missing from general relativity? While on the subject of arbitrarycurved manifolds, the idea of translational invariance is irrelevant, whence it follows thatnot only energy and momentum are not conserved, but also the very construction ofenergy and momentum suggested by Noether’s first theorem is no longer defined.To avoid this conclusion, one normally turns to field-theoretic treatments of gravity.This is feasible if the gravitational field can be granted to be ‘sufficiently weak’, g µν = η µν + φ µν , (61)12here | φ µν | ≪
1. The quantity φ µν is thought of as a second-rank tensor field definedin a flat background R , whose symmetry properties enable us to endow the resultingdynamics with conserved energy-momentum through the standard Noether’s prescription.It is believed that general relativity leaves room for both weak and strong gravity.Strange though it may seem, a simple and convincing criterion for discriminating between weak and strong gravity still remains to be established. We therefore have to address thisissue. But our concern here is not with elaborating this criterion in every respect. Rather,we only state the central idea and explicate it by the example of the Schwarzschild metric.Intuition suggests that the strong gravity should be associated with a great warpingof spacetime. However, a characteristic curvature whereby the changes in spacetimeconfigurations might be rated as ‘drastic’ is absent from general relativity, sending us insearch of another measure of such changes. It seems reasonable to assume that switchingbetween weak and strong gravitational regimes is due to spacetime topology alterations.The ‘strong gravitational field’ is then recognized as a qualitative rather than quantitativeconcept. The field equations (59), being differential equations, are local in character.They tell nothing about the topology of their solutions. A global solution can be recoveredwhen its infinitesimal pseudoeuclidean fragments are assembled into an integral dynamicalpicture, and the topology of this solution may well differ from the topology of Minkowskispacetime if the assembly is subject to a restrictive boundary condition. To illustrate, werefer to the Schwarzschild metric [17], ds = − (cid:16) − r S r (cid:17) dt + (cid:16) − r S r (cid:17) − dr + r d Ω , (62)where, d Ω is the round metric in a sphere S , and r S = 2 G N M is the Schwarzschild radius.A 3-dimensional spacelike surface Σ endowed with this metric has a twofold geometricinterpretation. First, it looks like a ‘bridge’ between two otherwise Euclidean spaces, and,second, it may be regarded as the ‘throat of a wormhole’ connecting two distant regionsin one Euclidean space in the limit when this separation of the wormhole mouths is verylarge compared to the circumference of the throat [18].To describe a curved manifold M , a set of overlapping coordinate patches covering M is called for. If one yet attempts to use a single coordinate patch, a singularity in theresulting description can arise. The gravitation is amenable to a field-theoretic treatmentuntil the mapping of the metric g µν into the field φ µν , as shown in (61), is bijective andsmooth, which is the same as saying that every curved spacetime configuration, associatedwith some gravitational effect, can be smoothly covered with a single coordinate patch.In contrast, for a manifold whose topology is nontrivial, the quest for a single-patchcovering culminates in a singular boundary, bearing some resemblance to a shock wave,as exemplified by the Schwarzschild metric (62) in which the coefficient of dr becomessingular at r = r S , so that this solution exhibits a standing spherical shock wave of thegravitational field.One may argue that this is an apparent singularity, related to the choice of coordinates,because the curvature invariants are finite and well behaved at r = r S , and, furthermore,the equation for the geodesics (58) shows a singular behavior only at r = 0. There arecoordinates u and v , proposed in [19] and [20], u = r rr S − (cid:18) rr S (cid:19) cosh (cid:18) tr S (cid:19) , v = r rr S − (cid:18) rr S (cid:19) sinh (cid:18) tr S (cid:19) , (63) Recent developments in bimetric theories of gravitation is reviewed in [16]. u and v , is regular in thewhole ( u, v ) plane, except for the point v − u = 1 corresponding to r = 0.In response to this objection, we would note that the introduction of these u and v isa clever trick to drive the shock wave in the singular point r = 0. However, our primeinterest is with the very existence of a shock wave, as evidence of the nontrivial topology,rather than its position in a particular coordinate system. The apparent regularity ofthe metric everywhere except r = 0 is due to an unfortunate choice of coordinates whichhides the Schwarzschild shock wave.The ‘floating’ position of the shock wave makes it clear that the strong gravitationalregime is unrelated to the magnitude of field variables. It is a topologically nontrivialaffair which renders the regime strong.This brings up the question as to whether the violation of regular behavior of theSchwarzschild metric at r = r S is an artefact of the original Schwarzschild description.Some fifty years ago people were inclined to believe that such is indeed the case. Bynow, however, r S is recognized as an objectively existing entity to characterize the eventhorizon of an isolated spherically symmetric stationary black hole. The event horizon ofa Schwarzschild black hole shows a demarcation between spacetime regions characterizedby opposite signatures of the metric . This geometrical layout, if it exists, provides anexplicit scheme for interfacing the classical and the quantum [21].Let us take a closer look at why energy and momentum are to be regarded as poorlydefined concepts. General relativity allows the Hamiltonian formulation for at least suchsystems whose geometric rendition is compatible with the idea of asymptotically flatspacetime [22]–[24]. More specifically, one supposes that the metric g µν approaches theLorentz metric η µν at spatial infinity sufficiently rapidly, namely g µν = η µν + O (cid:18) r (cid:19) , ∂ λ g µν = O (cid:18) r (cid:19) , r → ∞ . (64)The second condition is claimed to be needed so that the Lagrangian L = Z d x L ( t, x ) (65)with the commonly used first order Lagrangian density of the gravitational field sector L = √− g g µν (cid:0) Γ σµν Γ λσλ − Γ λµσ Γ σνλ (cid:1) (66)should converge. The volume integral in (65) diverges for the Schwarzschild solutionexpressed in terms of the original Schwarzschild coordinates, appearing in (62), because L = O (1) as r → ∞ . In contrast, the use of isotropic coordinates, which recasts theSchwarzschild metric (62) into ds = − − r S r r S r dt + (cid:16) r S r (cid:17) (cid:0) dx + dy + dz (cid:1) , (67)results in L = O (1 /r ), and hence affords the convergence of the Lagrangian (65). Itfollows from this simple example that both the asymptotical flatness R αβγδ → , r → ∞ , (68) Note that the only quantity which has a discontinuity jump at the front of a strong gravitationalshock wave is the signature because the metric immediately anterior and posterior to the front can bebrought to either of two diagonal forms: diag(+ − −− ) or diag( − + ++). P µ has an unambiguous significance. We now examine thecorrectness of this expectation restricting ourselves to P for simplicity.The total energy is given by the numerical value of the Hamiltonian E = Z d x H ( t, x ) . (69) H is a cumbersome construction which is immaterial for our discussion. However, the keypart of this construction proves to be cast [25] in a convenient form, E = 116 π I dS j (cid:18) ∂∂x i g ij − ∂∂x j g ii (cid:19) . (70)Here, the integral is evaluated over a 2-dimensional surface at spatial infinity.It is possible to prove [26, 27] that an isolated gravitating system having non-negativelocal mass density has non-negative total energy E . For example, for the Schwarzschildconfiguration generated by a point particle of mass m the surface integral (70) is easilyevaluated to give E = m. (71)Could the condition (64) be relaxed so that the asymptotical flatness condition (68)would hold, and every pertinent additive quantity in this Hamiltonian formulation remainsconvergent? To be more precise, we proceed from the metric g µν exhibiting the asymptoticbehavior (64), and transform the initial spatial coordinates x i into new ones ¯ x i , x i = ¯ x i [1 + f (¯ r )] , (72)where f is an arbitrary regular function subject to the following conditions: f (¯ r ) ≥ , lim ¯ r →∞ f (¯ r ) = 0 , lim ¯ r →∞ ¯ r f ′ (¯ r ) = 0 . (73)For the mapping (72) to be bijective, the condition ∂r∂ ¯ r = 1 + f (¯ r ) + ¯ rf ′ (¯ r ) > J = det (cid:18) ∂x∂ ¯ x (cid:19) = [1 + f (¯ r )] ∂r∂ ¯ r = 0 . (75)One such example [28] is f (¯ r ) = 2 α r l ¯ r (cid:20) − exp (cid:18) − ǫ ¯ rl (cid:19)(cid:21) , (76)where α and ǫ are arbitrary nonzero numbers, and l is an arbitrary parameter of dimensionof length. This is a bijective monotonically increasing regular mapping r → ¯ r which15ecomes 1 as ǫ →
0. The leading asymptotical terms of spatial components of the metricand those of the Christoffel symbols can be shown [28] to be g ij = δ ij + O (cid:18) r / (cid:19) , Γ ijk = O (cid:18) r / (cid:19) , ¯ r → ∞ , (77)while the Lagrangian density behaves as L = O (cid:18) r / (cid:19) , ¯ r → ∞ . (78)This provides the convergence of the volume integral in (65). Other additive quantitiesprove to be convergent as well .The mapping (72) with f defined in (76) is instructive to apply to the Schwarzschildmetric which is initially written in terms of isotropic coordinates (67). One can show [28]that the total energy of the Schwarzschild configuration generated by a point particle ofmass m takes any positive values, greater than, or equal to m , when α runs through R + , E = m (cid:0) α (cid:1) . (79)We thus see that the total energy of gravitational systems with nontrivial topologicalcontents depends on the foliation of spacetime. The Schwarzschild solution expressedin terms of coordinates for which the asymptotic condition is given in a relaxed form,Eq. (77), is a good case in point. This solution rearranges the initial degrees of freedomappearing in the Lagrangian density (66) to yield a coordinatization-dependent expressionfor the total energy functional (70). The same is true of the associated momentum.The situation closely parallels that in the paradoxial Banach–Tarski theorem whichstates [29]: given a unit ball in three dimensions, there exists a decomposition of this ballinto a finite disjoint subsets which can then be reassembled through continuous movementsof the pieces, without running into one another and without changing their shape, toyield another ball of larger radius. These situations share a common trait in that boththe Banach–Tarski decomposition and the Schwarzschild black hole formation are dueto topological rearrangements which are responsible for making the three-dimensionalmeasures of the resulting geometrical layouts poorly defined. The measure appearing inthe Banach–Tarski theorem is the ordinary volume of the balls (more precisely, Lebesguemeasure), while the measure in the gravitational energy problem is that of the functional(69). When turning to the surface integral for calculation of the total energy, Eq. (70),there arises the situation which may be likened to that of the Hausdorff paradox onenlarging spheres [30].The usual inference that the Banach–Tarski partitioning procedure has nothing todo with physical reality because there is no material ball which is not made of atomsoverlooks one important instance – black holes. Each isolated, stationary black hole iscompletely specified by three parameters: its mass m , angular momentum J , and electriccharge e . Whatever the content of a system which collapses under its own gravitationalfield, the exterior of the resulting black hole is described by a Kerr–Newman solution. Allinitial geometric features of the collapsing system, except for those peculiar to a perfectball, which may possibly rotate and carry electric charge, disappear in the black hole state[31, 32]. Furthermore, the event horizon which is meant for personifying the black hole isdevoid of the grain structure that was inherent in the collapsing system. Note also that the asymptotical flatness, Eq. (68), is still the case. Discussion and outlook
In Sects. 2 and 3 we saw that a careful analysis of the self-interaction problem may givean insight into the relation between the action–reaction principle and energy-momentumconservation provided that the rearrangement of degrees of freedom is taken into account.Umezawa [33] was the first to put the term ‘rearrangement’ in circulation by the exampleof spontaneous symmetry breaking. The mechanism for rearranging classical gauge fieldswas further studied in [13], [11], [3]. While a precise formulation of this mechanism isstill an open problem, the intuitive idea underlying the rearrangement is quite simple.In choosing variables for the description of a field system, preference is normally givento those which are best suited for realizing all supposed fundamental symmetries of theaction. But some degrees of freedom so introduced may be unstable. This gives rise toreassembling the initial degrees of freedom into new, stable aggregates whose dynamics isinvariant under broken or deformed groups of symmetries. Aggregates obeying the usualrequirement of stability δS = 0 , δ S > ε λ = 0 and E µ = 0, Eqs. (4) and (3), together with the retardedboundary condition. However, this dynamics blows up on the world line, which, in viewof Eq. (5), is tantamount to stating that the measure T µν dσ ν is ill defined. It would betempting to construe such divergences as evidences of instability.We then divide T µν dσ ν into a well-defined part and the remainder. But this separationis ambiguous: an arbitrary regular term can be added to one part and subtracted fromthe other to give an equivalent separation. To fix the separation, we impose the conditionthat every term obey the local conservation law (21). The functionals (22) and (23),expressing, respectively, the four-momentum radiated by the charge and four-momentumextracted from a free field, refer to the well-defined part of T µν dσ ν . We complete thedefinition of (Θ µν bound + t µν ) dσ ν by carrying out the renormalization of mass, Eq. (11).The functional (26) is regarded as the four-momentum of a dressed charged particle. Asmight be expected, the rearrangement outcome, the Lorentz–Dirac equation Λ µ = 0,Eq. (12), governing the behavior of the dressed particle, is depleted of some symmetriesembedded in the action. Indeed, this dynamical equation is not invariant under timereversal s → − s .We thus come to a new on-shell dynamics in which the equation of motion for a bareparticle ε µ = 0 is replaced with the equation of motion for a dressed particle Λ µ = 0,and all relevant integral quantities are well defined. Therefore, the rearrangement of theMaxwell–Lorentz electrodynamics can be briefly outlined as follows: since the on-shelldynamics which arises from extremizing the action and imposing the retarded boundarycondition is divergent, the initial degrees of freedom appearing in the action are inducedto reassemble into new aggregates governed by a tractable dynamics. What are the waysopen to this reassembly? 17he arena for rearranging the Maxwell–Lorentz electrodynamics is a line R coveredby the evolution variable τ which parametrizes the world line, and a plane E spanned bytwo vectors R µ and V µ used in defining the retarded Li´enard–Wiechert 2-form F , Eq. (9).Reparametrization invariance of the action and local SL(2 , R ) invariance of the 2-form F control the rearrangement scenario. Hence, the ways open to the reassembly are specifiedby the properties of the local translation group T responsible for reparametrizations andthe local SL(2 , R ) group acting in the retarded field plane.Recall the main implication of reparametrization invariance, Noether’s second theorem[2]. It is convenient to restrict our consideration to an infinitesimal reparametrization δτ = ǫ ( τ ) , (81)where ǫ is an arbitrary smooth function of τ close to zero, which becomes vanishing atthe end points of integration. Variation of τ implies the corresponding variation of theworld line coordinates δz µ = ˙ z µ ǫ . (82)In response to the reparametrization (81)–(82), the action varies as δS = Z dτ ε µ ˙ z µ ǫ . (83)Let S be invariant under reparametrizations, δS = 0. Because ǫ is assumed to be anarbitrary function τ , one concludes that˙ z µ ε µ = 0 . (84)This equation is a manifestation of Noether’s second theorem: invariance of the actionunder the transformation group (81) involving an arbitrary infinitesimal function ǫ impliesa linear relation between Eulerians.The identity (84) suggests that ε µ contains the projection operator on a hyperplanewith normal ˙ z µ , ˙ z ⊥ µν = η µν − ˙ z µ ˙ z ν ˙ z , (85)annihilating identically any vector parallel to ˙ z µ . Reparametrization invariance bearson the projection structure of the basic dynamical law for a bare particle which can bewritten as ˙ z ⊥ ( ˙ p − f ) = 0 , (86)where p is the four-momentum of a bare particle, and f an external four-force.In view of the identities ˙ z = 1 , ˙ z · ¨ z = 0 , ˙ z · ... z = − ¨ z , the Lorentz–Dirac equation(12) can be brought to the form of Eq. (86) in which p is the four-momentum of a dressedparticle, defined in (26), and f is again an external four-force.The structure of (86) makes it clear that a dressed particle experiences only an externalforce. This equation contains no term through which the dressed particle interacts withitself. The rearrangement eliminates self-interaction. The rearranged dynamical picturecontains only autonomous, foreign to each other entities. The projector ˙ z ⊥ may arise in (86) from a completely different origin, namely smooth embedding ofNewtonian dynamics into sections of Minkowski space perpendicular to the world line [3].
18t may be worth pointing out that both equation of motion for a bare particle ε µ = 0and equation of motion for a dressed particle Λ µ = 0 are generally not invariant underreparametrizations. Instead, this local symmetry leaves its imprint on the form of ε µ andΛ µ through the presence of the projector ˙ z ⊥ .Invariance under the SL(2 , R ) group stems from the fact that the 2-form F describingthe retarded field of a single charge is proportional to R ∧ V , that is, F is decomposable[13, 3]. Given a decomposable 2-form F , the invariant P = F µν ∗ F µν is identically zero.As for the invariant S = F µν F µν , using (9)–(10), we find S = − e /ρ . Therefore, asingle charge moving along an arbitrary timelike world line generates the retarded field F µν of electric type. In other words, whatever the motion of the charge, there is a Lorentzframe of reference, special for each point x µ , such that only electric field persists, moreprecisely, | E | = e/ρ and B = 0.Rewrite (9) as F = eρ ̟ , ̟ = R ∧ V . (87)A pictorial rendition of the bivector ̟ is the parallelogram of the vectors R µ and V µ .The area A of the parallelogram is A = s − V (cid:18) V ⊥ R (cid:19) = V · R = 1 . (88)The bivector ̟ is invariant under the special linear group of real unimodular 2 × , R ) which rotate and deform the initial parallelogram, converting it to parallelogramsof unit area belonging to the plane spanned by the vectors R µ and V µ . Therefore, ̟ isindependent of concrete directions and magnitudes of the constituent vectors R µ and V µ . ̟ depends only on the parallelogram’s orientation. The parallelogram can always be builtfrom a timelike unit vector e µ and a spacelike imaginary-unit vector e µ perpendicular to e µ , ̟ = e ∧ e . In fact, there are three different cases:(i) V > e µ = V µ √ V , e µ = √ V (cid:18) − R µ + V µ V (cid:19) , (89)(ii) V < e µ = √− V (cid:18) R µ − V µ V (cid:19) , e µ = V µ √− V , (90)(iii) V = 0, e µ = 1 √ (cid:18) ρV µ + R µ ρ (cid:19) , e µ = 1 √ (cid:18) ρV µ − R µ ρ (cid:19) . (91)In the Lorentz frame with the time axis parallel to the vector e µ , all components of F µν are vanishing, except for F . The formulas (89)–(91) specify explicitly a frame in whichthe retarded electromagnetic field generated by a single arbitrarily moving charge appearsas a pure Coulomb field at each observation point. With a curved world line, this frameis noninertial.The decomposable 2-form F is invariant under the SL(2 , R ) transformations which canbe carried out independently at any spacetime point. Therefore, we are dealing with localinvariance. This invariance is not pertinent to electrodynamics as a whole, and hence19ives rise to no Noether identities. Rather, this is a property of the retarded solution F µν ret ,shown in Eq. (9)–(10) .Therefore, the retarded solution F µν ret is determined not only by the field as such butalso by the frame of reference in which this quantity is measured. On the other hand,Θ µν is not invariant under such SL(2 , R ) transformations; Θ µν carries information aboutboth the field and the Lorentz frame which is used to describe F µν ret . Nevertheless, thefunctionals (22), (23), and (26) are well defined and frame-independent.The rearrangement in the Yang–Mills–Wong theory shows a general resemblance ofthat in the Maxwell–Lorentz electrodynamics. The field strength generated by a singlequark is also given by a decomposable 2-form F in both Abelian and non-Abelian regimes.The retarded Yang–Mills field F is always invariant under the local group SL(2 , R ).A special feature of the Yang–Mills–Wong theory (as opposed to the Maxwell–Lorentzelectrodynamics) is that non-Abelian regimes of evolution exhibit spontaneously deformed gauge symmetries [11, 3]. Without going into detail, we explicate this phenomenon by thesimplest example. Consider the solution (41) which describes the retarded non-Abelianfield generated by a single quark in the SU(2) Yang–Mills–Wong theory. By introducingan alternative matrix basis T = T , T = iT , T = T , (92)we convert this solution to the form A µ = A aµ T a where all coefficients A aµ are imaginary.Elements of this basis obey the commutation relations of the sl(2 , R ) Lie algebra. We thussee that the gauge group of the solution (41) is actually SL(2 , R ) . Where does this groupof symmetry come from? Its origin bears no relation to spontaneous symmetry breakdown:SU(2) and SL(2 , R ) are the compact and noncompact real forms of the complex groupSL(2 , C ). Invariance of the action under SU(2) automatically entails its invariance underthe complexification of this group, SL(2 , C ). The emergence of a solution invariant undera real form of SL(2 , C ) different from the initial SU(2) is a rearrangement phenomenonspecific to the Yang–Mills–Wong theory, called spontaneous symmetry deformation. Thesolutions (41) and (40) are different not only in their symmetry aspect, but also in physicalmanifestations, say, the former manifests itself as the Yang–Mills field of ‘magnetic’ typewhile the latter is the Yang–Mills field of ‘electric’ type. Dressed quark, associated withthese solutions, are governed by respectively equations of motion (55) and (12), bothbeing in agreement with the action–reaction principle.The rearrangement of general relativity is vastly different from that of the Maxwell–Lorentz electrodynamics and Yang–Mills–Wong theory. Indeed, the total stress-energytensor T µν is identical to the left-hand side of Eq. (59), that is, the on-shell T µν is zero.It is therefore impossible to define a three-dimensional measure weighted with T µν . Andyet, the on-shell dynamics exhibits a kind of blow-up: gravitational singularities. Thistroublesome feature of the theory is found even if delta-function sources are substituted bycontinuously distributed matter obeying a reasonable energy condition, the local positiveenergy condition [34, 35, 36]. However, the responsibility for the rearrangement doesnot rest with the divergent dynamics. Gravitational degrees of freedom are induced to The advanced field F µν adv can also be represented in a form similar to (9)–(10), that is, the 2-form F adv is decomposable whereas combinations F ret + α F adv are not. This SL(2 , R ) gauge group should not be confused with the SL(2 , R ) symmetry transformations whichleave a decomposable 2-form F unchanged. Given the initial SU( N ) gauge symmetry with N ≥
2, thespontaneously deformed gauge symmetry is found to be embedded in the SL( N , R ) group [11, 3]. x µ = F µ ( x ′ ) , g µν ( x ) = ∂x ′ α ∂x µ ∂x ′ β ∂x ν g ′ αβ ( x ′ ) , (93)where F µ are arbitrary smooth functions. These transformations form an infinite group,the group of diffeomorphism invariance implying invariance of the metric under the localLorentz group and parallel transport group.It would be interesting to inquire into why the functionals (69) and (70) becomecoordinatization-dependent for systems having nontrivial topological contents in the lightof the analyses which are lumped together as the ‘Banach–Tarski theorem’ [37]. Notethat the very analogy between the Banach–Tarski decomposition and the rearrangementof gravitational degrees of freedom may seem in doubt because the former has to do withsets of points in Euclidean space E , whereas the latter refers to the pseudo-Riemannian metric structure. But the resemblance of these procedures is ensured by the fact that thestudy of the affair with E is actually transferred to exploring the properties of bijectivemappings of sets in E , and the like is true for the rearrangement of gravitational degreesof freedom. A central idea of the Banach–Tarski analyses is that if a bounded set canbe decomposed in a paradoxial way with respect to a group G , then G contains free subgroups, in particular a ball in E is SO(3)-paradoxial because the action of SO(3) isthat of a free non-Abelian isometry group [37]. The development of this idea in relationto the action of the isometry group composed of the local SO(1 ,
3) group and paralleltransport group, having free non-Abelian subgroups, may give a plausible explanationfor the fact that the measure of integral quantities such as (69) and (70) is to be poorlydefined.On the other hand, the Banach’s theorem stating that no paradoxial decompositionsexists in R and E [37] should be likened to the affair with the well-defined measuresin the rearranged Maxwell–Lorentz electrodynamics and Yang–Mills–Wong theory. Theclass of groups whose actions preserve finitely additive, isometry-invariant measures of thebounded sets on R and E are known to be amenable groups, specifically solvable groups,which include Abelian groups. It is conceivable that the groups of reparametrizations andlocal SL(2 , R ) transformations controlling the rearrangement of these theories have whatamounts to the desired properties of amenable groups.A natural question may now arise: What is the reason for the existence of scenariosin which gravitational degrees of freedom reassemble in a topologically nontrivial fashion,say, into a Schwarzschild black hole, so that the asymptotic condition (64) is met, and thetotal energy functional (70) becomes a well defined, non-negative quantity [26, 27]? Thesuggestion can be made that the diffeomorphisms controlling such scenarios are restrictedto the groups deprived of free subgroups.Does the action–reaction principle remain its validity for quantum field theories suchas quantum electrodynamics? Three obstacles apparently placed in incorporation of thisprinciple into the quantum context are as follows: • By virtue of vacuum polarization, the charge of a bare particle is no longer constant,but rather a time-varying dynamical quantity whose numerical value is determined byvirtual pair screening. It is unlikely that this fluctuating quantity may be taken to be a21easure of both variation of the electron state for a given electromagnetic field state andvariation of the state of electromagnetic field for a given electron state. • Heisenberg’s uncertainty principles is contrary to bringing a contact interaction intocoincidence with exact values of the four-momenta appearing in the local four-momentumbalance. In the quantum realm, the four-momentum balance is either nonlocal or fuzzy. • The rearrangement of initial degrees of freedom in the quantum picture occurs muchdifferently than in the classical picture. The criterion of stability, Eq. (80), is alien tothe quantum regime of evolution because any world line passing through the chosen endpoints – and not just the world line which renders the action extremal – contributes tothe Feynman path integral. Therefore, the instability is of little, if at all, significance forthe quantum rearrangement.However, it would be very strange if the Nature does reject the quantum utility of theprinciple which is so useful at the classical level.
References [1] Planck, M.: Bemerkungen zum Prinzip der Aktion Reaktion in der allgemeinen Dy-namik. Phys. Z. , 828-830 (1908).[2] Noether, E.: Invariante Variationsprobleme. Nachr. K¨on. Ges. der Wiss. G¨ottingen.Math.-Phys. Kl., , 235-258 (1918).[3] Kosyakov, B.: Introduction to the Classical Theory of Particles and Fields. Springer,Berlin (2007).[4] Barut, A. O.: Electrodynamics in terms of retarded fields. Phys. Rev. D , 3335-3336 (1974).[5] Dirac, P. A. M.: Classical theory of radiating electron Proc. Roy. Soc. A , 1572-1582 (1970).[8] Kosyakov, B. P.: Self-accelerated universe. Intern. J. Mod. Phys. A , 2459-2464(2005).[9] Wong, S. K.: Field and particle equation for the classical Yang–Mills field and par-ticles with isotopic spin. Nuovo Cimento A , 689-694 (1970).[10] Balachandran, A., Borchardt, S., and Stern, A.: Lagrangian and Hamiltonian de-scription of Yang–Mills particles. Phys. Rev. D , 3247-3256 (1978).[11] Kosyakov, B. P.: Exact solutions in the Yang–Mills–Wong theory. Phys. Rev. D ,5032-5048 (1998). 2212] Kosyakov, B. P.: Field of arbitrarily moving colored charge. Theor. Math. Phys. ,632-635 (1991).[13] Kosyakov, B. P.: Radiation in electrodynamics and Yang-Mills theory. Sov. Phys.-Usp. , 135-142 (1992).[14] Kosyakov, B. P.: Classical Yang–Mills field generated by two colored point charges.Phys. Lett. B , 471-476 (1993).[15] Kosyakov, B. P.: Exact solutions of the Yang–Mills equations with the source in theform of two point color charges. Theor. Math. Phys. , 409-421 (1994).[16] Schmidt-May, A., and von Strauss, M.: Recent developments in bimetric theory. J.Phys. A , 183001 (2016).[17] Schwarzschild, K.: ¨Uber das Gravitationsfeld eines Massenpunktes nach der Ein-stein’schen Theorie. Sitzber. K¨on. Preuss. Akad. Wiss. Berlin. Math.-Phys. Kl. ,189-196 (1916).[18] Fuller, R. W. and Wheeler, J. A.: Causality and multiply connected space-time.Phys. Rev. , 286-318 (1974).[26] Schoen, R., and Yau, S.-T.: On the proof of the positive mass conjecture in generalrelativity. Commun. Math. Phys. , 45-76 (1979).[27] Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. ,381-402 (1981).[28] Denisov, V. I., Solov’ev, V. O.: The energy determined in general relativity on thebasis of the traditional Hamilton approach does not have physical meaning. Theor.Math. Phys. , 832-841 (1983). 2329] Banach, S., Tarski, A.: Sur la d´ecomposition des ensembles de points en partiesrespectivement congruentes. Fundamenta Mathematicae , 61 (2002).[33] Umezawa, H.: Dynamical rearrangement of symmetries. I. The Nambu–Heisenbergmodel, Nuovo Cimento A , 450-475 (1965).[34] Penrose, R.: Gravitational collapse and space-time singularities. Phys. Rev. Lett.314